where \(v_0\)  is the constant velocity.
For the AOUP model, the force \({\bf f}\)  evolves according to \(\tau \dot{\bf f} = -{\bf f} + \eta _f \mu ^{-1}\) 
[1]}, [2]}, where \(\eta _f\)  is a delta correlated Gaussian noise,
\(\langle \eta _f(t) \cdot \eta _f(t^{\prime })\rangle = 2n D_f \delta (t-t^{\prime })\) , and \(D_{f}\)  is the diffusion constant of that process.
For this model
\(\langle f^2\rangle = \frac{n D_f}{\tau \mu ^2} , ~~~~ \text{AOUP}.\) 
